Abstract. We study certain families of oscillatory integrals I ϕ (a), parametrised by phase functions ϕ and amplitude functions a globally defined on R d , which give rise to tempered distributions, avoiding the standard homogeneity requirement on the phase function. The singularities of I ϕ (a) are described both from the point of view of the lack of smoothness as well as with respect to the decay at infinity. In particular, the latter will depend on a version of the set of stationary points of ϕ, including elements lying at the boundary of the radial compactification of R d . As applications, we consider some properties of the two-point function of a free, massive, scalar relativistic field and of classes of global Fourier integral operators on R d , with the latter defined in terms of kernels of the form I ϕ (a).

We introduce a global wave front set suitable for the analysis of tempered ultradistributions of quasianalytic Gelfand-Shilov type. We study the transformation properties of the wave front set and use them to give microlocal existence results for pullbacks and products. We further study quasianalytic microlocality for classes of localization and ultradifferential operators, and prove microellipticity for differential operators with polynomial coefficients.

We prove that Hörmander's global wave front set and Nakamura's homogeneous wave front set of a tempered distribution coincide. In addition we construct a tempered distribution with a given wave front set, and we develop a pseudodifferential calculus adapted to Nakamura's homogeneous wave front set.

We continue our study of tempered oscillatory integrals I ϕ (a), here investigating the link with a suitable symplectic structure at infinity, which we describe in detail. We prove adapted versions of the classical theorems, which show that tempered distributions of the type I ϕ (a) are indeed linked to suitable Lagrangians extending to infinity, that is, extending up to the boundary and in particular the corners of a compactification of T * R d to B d ×B d . In particular, we show that such Lagrangians can always be parametrized by non-homogeneous, regular phase functions, globally defined on some R d × R s . We also state how two such phase functions parametrizing the same Lagrangian may be considered equivalent up to infinity.

We characterize the Schwartz kernels of pseudodifferential operators of Shubin type by means of an FBI transform. Based on this we introduce as a generalization a new class of tempered distributions called Shubin conormal distributions. We study their transformation behavior, normal forms and microlocal properties.2010 Mathematics Subject Classification. 46F05,46F12,35A18,35A22.